Sprinkle all sides of the fish fillets with the salt.
Melt 2 tablespoons of the butter in a stainless skillet over medium-high heat. Add the fish and sear until it is opaque halfway through, 4 to 5 minutes. Add the remaining 2 tablespoons of butter to the skillet and, once it has melted, flip the fillets and sear on the second side until completely cooked through and the fillets slightly resist semi-firm pressure when pressed, another 3 to 4 minutes.
Transfer the fish to plates. Add the capers to the hot skillet and let them sizzle for 15 seconds (if they don’t sizzle, place the skillet over medium-high heat for 30 seconds or until they start to sizzle). Squeeze the lemon juice over the capers, sprinkle with the parsley, and stir to combine. Drizzle the lemon-caper butter over the fillets and serve immediately.
If the butter goes from dark brown to black while the fish is cooking, then after plating the fish drain the dark butter off and add a fresh tablespoon of butter to the pan. It will melt quickly. Then proceed with sauce as described above.